Tag - Dirichlet L-functions

Paul Bourgade: Random matrices, the Riemann zeta function and branching processes I

8th April, 2023

Random matrix theory is a powerful tool for prediction in analytic number theory. Through this random matrix analogy, Fyodorov, Hiary and Keating conjectured very precisely the typical values of the Riemann zeta function in short intervals of the critical line, in particular their maximum. Their prediction relied on techniques from statistical mechanics such as the replica method, giving extreme values in disordered systems. Recent rigorous progress has exploited underlying branching structures instead, both for random characteristic polynomials and L-functions.

Simon Marshall: Subconvexity for L-functions on U(n) x U(n+1)

6th April, 2023

We prove this bound by first using the unitary Ichino-Ikeda formula of N. Harris to relate the central L-value to an automorphic period integral. There is a `trivial' bound for this integral, which turns out to correspond to the convexity bound for the L-value if the test vector is chosen correctly. We are able to improve the bound for the period integral using a technique called arithmetic amplification, which uses the action of the Hecke operators, and this yields a subconvex bound.

Alexandre de Faveri: Simple Zeros of Modular Form L-Functions

3rd April, 2023

Zeros of L-functions have been extensively studied, due to their close connection to arithmetic problems. Despite several precise conjectures about their behaviour, our unconditional understanding of them remains limited. In this talk we will discuss certain intrinsic properties of such zeros, focusing on what is known (in degrees 1 and 2) about their accumulation on the central line and their multiplicity. Here the tools of analytic number theory can give quantitative advances, and we will show how to deduce that there are many zeros of multiplicity one for the L-function associated to a modular form.

Emanuel Carneiro: Hilbert spaces and low-lying zeros of L-functions

10th November, 2022

In this talk I would like to present some ideas behind a general Hilbert space framework for solving certain optimization problems that arise when studying the distribution of the low-lying zeros of families of L-functions. For instance, in connection to previous work of Iwaniec, Luo, and Sarnak (2000), we will discuss how to use information from one-level density theorems to estimate the proportion of non-vanishing of L-functions in a family at a low-lying height on the critical line. We will also discuss the problem of estimating the height of the first low-lying zero in a family, considered by Hughes and Rudnick (2003) and Bernard (2015). This is based on joint work with M. Milinovich and A. Chirre.

Jeffrey C. Lagarias: The Alternative Hypothesis and Point Processes

6th October, 2022

The Alternative Hypothesis concerns a hypothetical and unlikely picture of how zeros of the Riemann zeta function are spaced. It asks that nearly all normalized zero spacings be near half-integers. This possible zero distribution is incompatible with the GUE distribution of zero spacings. Ruling it out arose as an obstacle to the long-standing problem of proving there are no exceptional zeros of Dirichlet L-functions. The talk describes joint work with Brad Rodgers, that constructs a point process realizing Alternative Hypothesis type statistics, which is consistent with the known results on correlation functions for spacings of zeta zeros. (A similar result was independently obtained by Tao with slightly different methods.) The talk reviews point process models and presents further results on the general problem of to what extent two point processes, a continuous one on the real line, the other a discrete one on a lattice aZ, can mimic each other in the sense of having perfect agreement of all their correlation functions when convolved with bandlimited test functions of a given bandwidth B.

Adam Harper: Low moments of character sums

8th April, 2021

Sums of Dirichlet characters ∑n≤xχ(n) (where χ is a character modulo some prime r, say) are one of the best-studied objects in analytic number theory. Their size is the subject of numerous results and conjectures, such as the Pólya-Vinogradov inequality and the Burgess bound. One way to get information about this is to study the power moments 1/(r−1) ∑χ mod r|∑n≤xχ(n)|2q, which turns out to be quite a subtle question that connects with issues in probability and physics. In this talk I will describe an upper bound for these moments when 0≤q≤1. I will focus mainly on the number-theoretic issues arising.

Gal Dor: Monoidal Structures on GL2-Modules and Abstractly Automorphic Representations

4th March, 2021

Consider the function field F of a smooth curve over 𝔽q, with q>2. L-functions of automorphic representations of GL2 over F are important objects for studying the arithmetic properties of the field F. Unfortunately, they can be defined in two different ways: one by Godement-Jacquet, and one by Jacquet-Langlands. Classically, one shows that the resulting L-functions coincide using a complicated computation. Each of these L-functions is the GCD of a family of zeta integrals associated to test data. I will categorify the question, by showing that there is a correspondence between the two families of zeta integrals, instead of just their L-functions. The resulting comparison of test data will induce an exotic symmetric monoidal structure on the category of representations of GL2. It turns out that an appropriate space of automorphic functions is a commutative algebra with respect to this symmetric monoidal structure. I will outline this construction, and show how it can be used to construct a category of automorphic representations.

Joni Teräväinen: On the Liouville function at polynomial arguments

10th December, 2020

Let λ be the Liouville function and P(x) any polynomial that is not a square. An open problem formulated by Chowla and others asks to show that the sequence λ(P(n)) changes sign infinitely often. We present a solution to this problem for new classes of polynomials P, including any product of linear factors or any product of quadratic factors of a certain type. The proofs also establish some nontrivial cancellation in Chowla and Elliott type correlation averages.

Jingwei Xiao: A Unitary Analogue of Friedberg-Jacquet Periods and Central Values of Standard L Functions on GL(2n)

3rd December, 2020

Let G be a reductive group over a number field F and H a subgroup. Automorphic periods study the integrals of cuspidal automorphic forms on G over H(F)\H(AF). They are often related to special values of certain L-functions. One of the most notable cases is when (G,H)=(U(n+1)☓U(n), U(n)), and these periods are related to central values of Rankin-Selberg L-functions on GL(n+1)☓GL(n). In this talk, I will explain my work in progress with Wei Zhang that studies central values of standard L-functions on GL(2n) using (G,H)=(U(2n), U(n)☓U(n)) and some variants. I shall explain the conjecture and a relative trace formula approach to study it. We prove the required fundamental lemma using a limit of the Jacquet-Rallis fundamental lemma and Hironaka’s characterization of spherical functions on the space of non-degenerate Hermitian matrices. Also, the question admits an arithmetic analogue.